Lecture 13: Heisenberg and Uncertainty

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Lecture 13 Heisenberg and Uncertainty
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Determinism of Classical Mechanics

• Suppose the positions and speeds of all particles
  in the universe are measured to sufficient
  accuracy at a particular instant in time
• It is possible to predict the motions of every
  particle at any time in the future (or in the
  past for that matter)

An intelligent being knowing, at a given instant
of time, all forcesacting in nature, as well as
the momentary positions of all things ofwhich
the universe consists, would be able to
comprehend themotions of the largest bodies of
the world and those of the smallestatoms in one
single formula, provided it were sufficiently
powerfulto subject all the data to analysis to
it, nothing would be uncertain,both future and
past would be present before its eyes. Pierre
Simon Laplace
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Role of an Observer

• The observer is objective and passive
• Physical events happen independently of whether
  there is an observer or not
• This is known as objective reality

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Double-Slit Experimentcannot predict where
electron would land
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Double-Slit Experimentact of observation
affects behaviour of electron
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Role of an Observer in Quantum Mechanics

• The observer is not objective and passive
• The act of observation changes the physical
  system irrevocably
• This is known as subjective reality

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Heisenberg realised that ...

• In the world of very small particles, one cannot
  measure any property of a particle without
  interacting with it in some way
• This introduces an unavoidable uncertainty into
  the result
• One can never measure all the properties exactly

Werner Heisenberg (1901-1976)
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Measuring the position and momentum of an
electron

• Shine light on electron and detect reflected
  light using a microscope
• Minimum uncertainty in position is given by the
  wavelength of the light
• So to determine the position accurately, it is
  necessary to use light with a short wavelength

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Measuring the position and momentum of an
electron (contd)

• By Plancks law E hc/l, a photon with a short
  wavelength has a large energy
• Thus, it would impart a large kick to the
  electron
• But to determine its momentum accurately,
  electron must only be given a small kick
• This means using light of long wavelength!

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Fundamental Trade Off

• Use light with short wavelength
• accurate measurement of position but not momentum
• Use light with long wavelength
• accurate measurement of momentum but not position

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Heisenbergs Uncertainty Principle
The more accurately you know the position (i.e.,
the smaller Dx is) , the less accurately you
know the momentum (i.e., the larger Dp is) and
vice versa
applet
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Implications

• It is impossible to know both the position and
  momentum exactly, i.e., Dx0 and Dp0
• These uncertainties are inherent in the physical
  world and have nothing to do with the skill of
  the observer
• Because h is so small, these uncertainties are
  not observable in normal everyday situations

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Example of Baseball

• A pitcher throws a 0.1-kg baseball at 40 m/s
• So momentum is 0.1 x 40 4 kg m/s
• Suppose the momentum is measured to an accuracy
  of 1 percent , i.e., Dp
  0.01 p 4 x 10-2 kg m/s

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Example of Baseball (contd)

• The uncertainty in position is then
• No wonder one does not observe the effects of the
  uncertainty principle in everyday life!

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Example of Electron

• Same situation, but baseball replaced by an
  electron which has mass 9.11 x 10-31 kg
• So momentum 3.6 x 10-29 kg m/s and its
  uncertainty 3.6 x 10-31 kg m/s
• The uncertainty in position is then

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If Plancks constant were much larger...
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Another Consequence of Heisenbergs Uncertainty
Principle

• A quantum particle can never be in a state of
  rest, as this would mean we know both its
  position and momentum precisely
• Thus, the carriage will be jiggling around
  thebottom of the valleyforever

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Heisenbergs Uncertainty Principle involving
energy and time

• The more accurately we know the energy of a body,
  the less accurately we know how long it
  possessed that energy
• The energy can be known with perfect precision
  (DE 0), only if the measurement is made over an
  infinite period of time (Dt 8)

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Summary Lessons from Heisenberg

• The idea of a perfectly predictable universe
  cannot be true
• There is no such thing as an ideal, objective
  observer